A Fractional Calculus of Variations for Multiple Integrals with Application to Vibrating String∗ Ricardo Almeida1 Agnieszka B. Malinowska1,2 [email protected] [email protected] Delfim F. M. Torres1 [email protected] 1Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal 2Faculty of Computer Science, Bia lystok University of Technology, 15-351 Bia lystok, Poland Abstract We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann-Liouville fractional derivatives and integrals in the sense of Jumarie. Main results provide fractional versions of the theorems of Green and Gauss, fractional Euler-Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string. PACS: 45.10.Db; 45.10.Hj; 02.30.Xx. Mathematics Subject Classification 2010: 49K10; 26A33; 26B20. Keywords: fractional calculus; modified Riemann-Liouville derivatives and integrals; fractional calculus of variations; multiple integrals; fractional theorems of Green, Gauss, and Stokes; fractional Euler-Lagrange equations. 1 Introduction arXiv:1001.2722v1 [math.OC] 15 Jan 2010 The Fractional Calculus (FC) is one of the most interdisciplinary fields of mathematics, with many applications in physics and engineering. The history of FC goes back more than three centuries, when in 1695 the derivative of order α =1/2 was described by Leibniz. Since then, many different forms of fractional operators were introduced: the Grunwald-Letnikov, Riemann-Liouville, Riesz, and Caputo fractional derivatives [31, 35, 39], and the more recent notions of Klimek [32], Cresson [14], and Jumarie [24, 25, 27, 28]. FC is nowadays the realm of physicists and mathematicians, who investigate the usefulness of such non-integer order derivatives and integrals in different areas of physics and mathematics [12, 23, 31]. It is a successful tool for describing complex quantum field dynamical systems, dissipation, and long-range phenomena that cannot be well illustrated using ordinary differential and integral operators [20, 23, 32, 38]. Applications of FC are found, e.g., in classical and quantum mechanics, field theories, variational calculus, and optimal control [19, 22, 27]. Although FC is an old mathematical discipline, the fractional vector calculus is at the very beginning. We mention the recent paper [42], where some fractional versions of the classical results of Green, Stokes and Gauss are obtained via Riemann-Liouville and Caputo fractional operators. ∗Accepted for publication in the Journal of Mathematical Physics (14/January/2010). 1 For the purposes of a multidimensional Fractional Calculus of Variations (FCV), the Jumarie fractional integral and derivative seems, however, to be more appropriate. The FCV started in 1996 with the work of Riewe [38]. Riewe formulated the problem of the calculus of variations with fractional derivatives and obtained the respective Euler-Lagrange equations, combining both conservative and nonconservative cases. Nowadays the FCV is a sub- ject under strong research. Different definitions for fractional derivatives and integrals are used, depending on the purpose under study. Investigations cover problems depending on Riemann- Liouville fractional derivatives (see, e.g., [6, 20, 21]), the Caputo fractional derivative (see, e.g., [1, 8, 9]), the symmetric fractional derivative (see, e.g., [32]), the Jumarie fractional derivative (see, e.g., [5, 27]), and others [3, 14, 19]. For applications of the fractional calculus of variations we refer the reader to [16, 17, 20, 27, 32, 36, 37, 40]. Although the literature of FCV is already vast, much remains to be done. Knowing the importance and relevance of multidimensional problems of the calculus of varia- tions in physics and engineering [43], it is at a first view surprising that a multidimensional FCV is a completely open research area. We are only aware of some preliminary results presented in [20], where it is claimed that an appropriate fractional variational theory involving multiple in- tegrals would have important consequences in mechanical problems involving dissipative systems with infinitely many degrees of freedom, but where a formal theory for that is missing. There is, however, a good reason for such omission in the literature: most of the best well-known fractional operators are not suitable for a generalization of the FCV to the multidimensional case, due to lack of good properties, e.g., an appropriate Leibniz rule. The main aim of the present work is to introduce a fractional calculus of variations for multiple integrals. For that we make use of the recent Jumarie fractional integral and derivative [24, 25, 28], extending such notions to the multidimensional case. The main advantage of using Jumarie’s approach lies in the following facts: the Leibniz rule for the Jumarie fractional derivative is equal to the standard one and, as we show, the fractional generalization of some fundamental multidimensional theorems of calculus is possible. We mention that Jumarie’s approach is also useful for the one-dimensional FCV, as recently shown in [5] (see also [26]). The plan of the paper is as follows. In Section 2 some basic formulas of Jumarie’s fractional calculus are briefly reviewed. Then, in Section 3, the differential and integral vector operators are introduced, and fractional Green’s, Gauss’s and Stokes’ theorems formulated. Section 4 is devoted to the study of problems of fractional calculus of variations with multiple integrals. Our main results provide Euler-Lagrange necessary optimality type conditions for such problems (Theorems 3 and 5) as well as natural boundary conditions (Theorem 4). We end with Section 5 of applications and future perspectives. 2 Preliminaries For an introduction to the classical fractional calculus we refer the reader to [31, 34, 35, 39]. In this section we briefly review the main notions and results from the recent fractional calculus proposed by Jumarie [25, 27, 28]. Let f : [0, 1] → R be a continuous function and α ∈ (0, 1). The Jumarie fractional derivative of f may be defined by x 1 d − f (α)(x)= (x − t) α(f(t) − f(0)) dt. (1) Γ(1 − α) dx Z0 One can obtain (1) as a consequence of a more basic definition, a local one, in terms of a fractional finite difference (cf. equation (2.2) of [28]): ∞ 1 α f (α)(x) = lim (−1)k f (x + (α − k)h) . h→0 hα k kX=0 Note that the Jumarie and the Riemann-Liouville fractional derivatives are equal if f(0) = 0. The advantage of definition (1) with respect to the classical definition of Riemann-Liouville is that the 2 fractional derivative of a constant is now zero, as desired. An anti-derivative of f, called the (dt)α integral of f, is defined by x x − f(t)(dt)α = α (x − t)α 1f(t)dt. Z0 Z0 The following equalities can be considered as fractional counterparts of the first and the second fundamental theorems of calculus and can be found in [27, 28]: α x d α α f(t)(dt) = α!f(x), (2) dx Z0 x f (α)(t)(dt)α = α!(f(x) − f(0)), (3) Z0 where α! := Γ(1 + α). The Leibniz rule for the Jumarie fractional derivative is equal to the standard one: (f(x)g(x))(α) = f (α)(x)g(x)+ f(x)g(α)(x). Here we see another advantage of derivative (1): the fractional derivative of a product is not an infinite sum, in opposite to the Leibniz rule for the Riemann-Liouville fractional derivative [35, p. 91]. One can easily generalize the previous definitions and results for functions with a domain [a,b]: x 1 d − f (α)(x)= (x − t) α(f(t) − f(a)) dt Γ(1 − α) dx Za and x x − f(t)(dt)α = α (x − t)α 1f(t)dt. Za Za 3 Fractional integral theorems In this section we introduce some useful fractional integral and fractional differential operators. With them we prove fractional versions of the integral theorems of Green, Gauss, and Stokes. Throughout the text we assume that all integrals and derivatives exist. 3.1 Fractional operators n Rn Let us consider a continuous function f = f (x1,...,xn) defined on R = Πi=1[ai,bi] ⊂ . Let us extend Jumarie’s fractional derivative and the (dt)α integral to functions with n variables. For xi ∈ [ai,bi], i =1,...,n, and α ∈ (0, 1), we define the fractional integral operator as xi α α−1 ai Ixi [i]= α (xi − t) dt . Zai These operators act on f in the following way: xi α α−1 − ai Ixi [i]f(x1,...,xn)= α f(x1,...,xi 1,t,xi+1,...,xn)(xi − t) dt, i =1, . , n . Zai Let Ξ = {k1,...,ks} be an arbitrary nonempty subset of {1,...,n}. We define the fractional s multiple integral operator over the region RΞ = Πi=1[aki , xki ] by α α α I [k ,...,ks]= a I [k ] ... a I [ks] RΞ 1 k1 xk1 1 ks xks xk1 xks s α−1 α−1 = α ··· (xk1 − tk1 ) ··· (xks − tks ) dtks ...dtk1 Zak1 Zaks 3 which acts on f by α IRΞ [k1,...,ks]f(x1,...,xn) xk1 xks s α−1 α−1 = α ··· f(ξ1,...,ξn)(xk1 − tk1 ) ··· (xks − tks ) dtks ...dtk1 , Zak1 Zaks where ξj = tj if j ∈ Ξ, and ξj = xj if j∈ / Ξ, j =1,...,n. The fractional volume integral of f over the whole domain R is given by b1 bn α n α−1 α−1 IRf = α ··· f(t1,...,tn)(b1 − t1) ··· (bn − tn) dtn ...dt1. Za1 Zan The fractional partial derivative operator with respect to the ith variable xi, i = 1,...,n, of order α ∈ (0, 1) is defined as follows: xi α 1 ∂ −α ai Dxi [i]= (xi − t) dt , Γ(1 − α) ∂xi Zai which act on f by α ai Dxi [i]f(x1,...,xn) xi 1 ∂ −α = (xi−t) [f(x1,...,xi−1,t,xi+1,...,xn) − f(x1,...,xi−1,ai, xi+1,...,xn)] dt , Γ(1 − α) ∂xi Zai i =1,...,n.